1

Uses both direction (sign) and magnitude.

Applies to the case of symmetric continuous distributions: Mean equals median.

Wilcoxon Signed-Rank Test

2

H0: = 0

Compute differences, Xi – 0 , i = 1,2,…,n

Rank the absolute differences | Xi – 0 | W+ = sum of positive ranks W– = sum of negative ranks From Table X in Appendix: critical w*

What are the rejection criteria for different H1?

Method

3

If sample size is large, n > 20 W+ (or W–) is approximately normal with

Large Samples

2

( 1)

4( 1)(2 1)

24

W

W

n n

n n n

4

Paired data has to be from two continuous distributions that differ only wrt their means.

Their distributions need not be symmetric.

This ensures that the distribution of the differences is continuous and symmetric.

Paired Observations

5

If underlying population is normal, t-test is best (has lowest ).

The Wilcoxon signed-rank test will never be much worse than the t-test, and in many nonnormal cases it may be superior.

The Wilcoxon signed-ran test is a useful alternate to the t-test.

Compare to t-test

6

Data from two samples with underlying distributions of same shape/spread, n1 n2

Rank all n1+n2 observations in ascending order

W1 = sum of ranks in sample 1

W2 = 0.5(n1+n2)(n1+n2+1) – W1

Table XI in the Appendix contain the critical value of the rank sums. What are the rejection criteria?

Wilcoxon Rank-Sum Test

7

If sample sizes are large, n1,n2 > 8

W1 is approximately normal with

Large Samples

1

1

1 1 2

2 1 2 1 2

( 1)

2( 1)

12

W

W

n n n

n n n n

8

When underlying distributions are normal, the Wilcoxon signed rank and rank-sum tests are approx 95% as efficient as the t-test in large samples.

Regardless of the distribution, the Wilcoxon tests are at least 86% as efficient.

Efficiency of Wilcoxon test relative to t-test is usually high if distributions have heavier tails than the normal.

Compare to t-test